How much gravity would our moon have if it was made out of solid gold? :hmm:
Using Orbiter's Moon radius we find the volume:
[math] V = \frac{4}{3}\pi R^3=\frac{4}{3}\pi \cdot(1.738\cdot 10^6)^3\simeq 2.199064\cdot 10^{19} \; m^3[/math]
Gold's density is [math] 19300 \; \frac{kg}{m^3}[/math]
So the mass of the Moon would be [math]19300 \cdot 2.199064\cdot 10^{19}=4.24419352\cdot 10^{23}\; kg[/math]
That's about 14 times less than the mass of the Earth and about 5.78 times more than the Moon's current mass.
The surface gravity is [math] g = \frac{GM}{R^2} = \frac{6.67259\cdot 10^{-11} \cdot 4.24419352\cdot 10^{23}}{(1.738\cdot 10^6)^2}\simeq 9.375 \; \frac{m}{s^2}\;\;\;[/math] which is about 96% of Earth's gravity.
That is not an accurate number of course, because it doesn't take into account the
gravitational compression. A Moon volume of gold wouldn't be made of
solid gold. The gold at the centre would be in liquid state, induced by the pressure of the rest of the gold around it. The liquid state has a larger volume. Larger volume = larger radius, thus the surface gravity would be smaller. (I don't think the gravity would be much smaller because this is an inverse cubed relationship).
-note to self #1: Learn more about gravitational compression and calculate the volume of the liquid state gold.
-note to self #2: Become a doctor and change last name to Acula.
More to the point, if you had a moon-mass of solid gold, how big would it be compared to the Moon.
Orbiter's Moon mass is [math]7.347664\cdot 10^{22} \; kg[/math]
With a gold density of [math] 19300\; {kg}{m^3}[/math] the volume would be:
[math] V = \frac{mass}{density}= \frac{7.347664\cdot 10^{22}}{19300}=3.8070797927461\cdot 10^{18} \; m^3[/math]
The radius is:
[math] V = \frac{4}{3}\pi R^3\Rightarrow R^3=\frac{V}{\frac{4}{3}\pi}\Rightarrow R=\left( \frac{V}{\frac{4}{3}\pi}\right)^{\frac{1}{3}}= \left( \frac{3.8070797927461\cdot 10^{18}}{\frac{4}{3}\pi}\right)^{\frac{1}{3}}\simeq0.968652\cdot 10^6\; m[/math] which is about 56% of the Moon's current radius. I don't think there's a question about such an object's roundness, because of it's mass.
One question that wasn't asked was "How much would that amount of bling bling bl*slap face* be worth?"
With today's prices (~1295 $/kg ,July 19 2013) a Moon's volume worth of gold would be at around [math]5.494\cdot 10^{26} dollars[/math] That's about 20 powers of ten more of the richest country's GDP *coughUSAcough*
Even in the case of a Moon-mass gold object, the value is 'only" one power of 10 lower.
Want to compare against the world's monetary output? You are probably more than 25 orders of magnitude inadequate...
